In the bowl of a stand mixer fitted with the paddle attachment, beat butter, brown sugar, salt, baking soda, and vanilla on medium-high speed until smooth and creamy, about 3 minutes. Add flour, and mix on low speed until just combined, about 1 minute. Using a flexible spatula, scrape bowl and beater to make sure there are no streaks of unmixed dough, then knead lightly against sides of bowl. Halve the dough; place 1 portion of dough back into bowl.

On a cutting board or work surface, shape dough into a 1 ½-inch wide log about 8 inches long and wrap tightly with plastic, twisting the ends of the plastic like a Tootsie Pop. Once wrapped, roll log under your hands in a back-and-forth motion to ensure log is smooth, even, and tight all around. Repeat with remaining dough. Chill until firm, about 1 hour. Line two 13- by 18-inch rimmed baking sheets with parchment; set aside.

Adjust oven rack to middle position and preheat to 325°F (160ºC). Remove plastic wrap and set dough logs on a clean cutting board. Using a sharp chef’s knife, slice each dough log into rounds just shy of ½ inch; you should have 16 cookies per log. Place granulated sugar in a wide bowl. Dip each cookie in granulated sugar to lightly coat, then set on prepared baking sheets, taking care to leave about an inch of space between each cookie. Sprinkle each cookie with a pinch of flaky salt.

Bake 1 sheet of cookies until they have very slightly puffed and are matte and dry to the touch, about 15 minutes. (The cookies may still feel quite soft but will firm up as they cool.) Repeat with remaining baking sheet.

Let cool for 5 to 10 minutes before serving. Enjoy warm or at room temperature.